Optimal investment is a tremendously rich source of mathematical
challenges in stochastic control theory. The key driver in this
problem is the tradeoff between risk and return and, thus, information
on the investment opportunities is playing a role which is as
important for the mathematical theory as it is in practice where
investors go at great lengths to secure even the slightest advantage
in knowledge. So it is no wonder that insider information has been
widely studied in the literature; see, for instance,
[1]}, [2]}, [3]}, [4]}, [5]} where an investor obtains extra
information about the stock price evolution at some point in time. By
contrast to these studies, the present paper takes a more dynamic view
on information gathering and affords the investor the opportunity to
continually peek \(\Delta \)  units of time into the future.
Closest to such an investor in reality may be high-frequency traders
that get access to or process order flow information faster than their
competition. To the best of our knowledge, this paper is the first
continuous-time stochastic control paper with such a feature, apart
from the optimal stopping problem of [6]}.
